Establishment of coverage-mass equation to quantify the corrosion inhomogeneity and examination of medium effects on iron corrosion

Abstract Metal corrosion is important in the fields of biomedicine as well as construction and transportation etc. While most corrosion occurs inhomogeneously, there is so far no satisfactory parameter to characterize corrosion inhomogeneity. Herein, we employ the Poisson raindrop question to model the corrosion process and derive an equation to relate corrosion coverage and corrosion mass. The resultant equation is named coverage-mass equation, abbreviated as C-M equation. We also suggest corrosion mass at 50% coverage, termed as half-coverage mass Mcorro50%, as an inhomogeneity parameter to quantify corrosion inhomogeneity. The equation is confirmed and the half-coverage mass Mcorro50% is justified in our experiments of iron corrosion in five aqueous media, normal saline, phosphate-buffered saline, Hank’s solution, deionized water and artificial seawater, where the former three ones are biomimetic and very important in studies of biomedical materials. The half-coverage mass Mcorro50% is proved to be more comprehensive and mathematically convergent than the traditional pitting factor. Iron corrosion is detected using visual observation, scanning electron microscopy with a build-in energy dispersive spectrometer, inductive coupled plasma emission spectrometry and electrochemical measurements. Both rates and inhomogeneity extents of iron corrosion are compared among the five aqueous media. The factors underlying the medium effects on corrosion rate and inhomogeneity are discussed and interpreted. Corrosion rates of iron in the five media differ about 7-fold, and half-coverage mass values differ about 300 000-fold. The fastest corrosion and the most significant inhomogeneity occur both in biomimetic media, but not the same one. The new equation (C-M equation) and the new quantity (half-coverage mass) are stimulating for dealing with a dynamic and stochastic process with global inhomogeneity including but not limited to metal corrosion. The findings are particularly meaningful for research and development of next-generation biodegradable materials.


Supplementary Formula Derivation
The corrosion process with time t is divided into two stages, nucleation and growth.
The pit nucleation is presumed to be generated in a homogeneous way with the kinetic ct or in a heterogeneous way of a constant N0. In both cases, the pit radii are assumed to grow with 1 1 and the average pit depths are assumed to grow with 2 2 .
The corrosion nucleation is regarded by us as a stochastic process and described by the Poisson raindrop question. As schematically presented in Figure 2 in the main manuscript, if the number of raindrop (or corrosion pit) passing through a random point Q is denoted as x, its probability P(x) must follow the Poisson distribution with the form of equation (2) in the main manuscript. The uncorroded area is thought to be the region with zero "raindrop wave". The corrosion coverage (θ) is thus expressed as equation (4), and could be acquired by calculation of λ, the average number of "raindrop waves" or corrosion pits passing through point Q.
We derive the equation first for the case of homogeneous nucleation. In a "ripple" ring with the distance L from point Q, only the corrosion pits whose generation time is larger than ( 1 ) 1 1 could pass through Q. So After integration, λ is expressed as Combination of this equation with equation (4) leads to equation (5) describing corrosion coverage (θ) versus corrosion time t.

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The relation between θ and t is thus similar to the Avrami equation with the form of equation (6).
Then we deal with corrosion mass per initial surface (Mcorro). In the case of homogeneous nucleation, the variation of pit density N(t) could be written as ∆ ( ) = ∆ .
The time t is related to r as So Δt between r ~ r + Δr is expressed as And the amount of the corrosion pits between r and r+Δr is written as The variation of the corrosion volume ΔVcorro is expressed as And the corrosion volume per unit area of iron could be expressed as The corrosion mass (Mcorro) per area of iron is thus obtained as shown in equation (7).
We eliminate the time t after combination of equations (5) and (7) and thus obtain equation (9), namely, the C-M equation.
Now we discuss the case of heterogeneous nucleation, which occurs, for instance, around some preformed defects. In this case, the nucleation number is a constant, denoted as N0. Then, the average number of raindrop waves or corrosion pits passing S4 through a random point Q at time t is integrated as After combining equation (4), corrosion coverage (θ) as a function of time is written as equation (14).
As for the corrosion volume and thus mass in the case of heterogeneous nucleation, all the corrosion pits are generated at the same time. So the corrosion volume per area of iron could be calculated by corrosion pit density multiply the volume of the single corrosion pits The corrosion mass (Mcorro) of the unit cm 2 area of the metal is thus written as equation (15).
Combination of equations (14) and (15) gives equation (9) again, and the corresponding k and n are expressed as equations (16) and (17), respectively. Hence, corrosion kinetics with either homogeneous nucleation or heterogeneous nucleation obeys our C-M equation.
It is necessary to indicate that overlapping among pits on the dimension of the metal surface has not been taken into consideration in derivation of corrosion volume and corrosion mass in both cases of homogeneous nucleation and heterogeneous nucleation.
Such an assumption could be regarded as compensated by another assumption of the power relation of the average pitting depth with corrosion time. In any case, the C-M equation stands, given a power relation of corrosion volume or mass as a function of corrosion time.

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According to our corrosion experiments, the iron corrosion proceeded like a homogeneous nucleation. It seems also helpful to indicate that the concrete expression forms of the k and n in the C-M equation (equation 9) like equations (12) and (13) for an ideal homogenous nucleation, equations (16) and (17) for an ideal heterogenous nucleation, or their mixing case do not influence the application of our new and unified inhomogeneity parameter Mcorro50%, namely, corrosion mass at 50% coverage simply called half-coverage mass. Figure S1. Publication statistics about "corrosion" during the latest 10 years.

Supplementary Figures and Tables
Source: Web of Science.

Subject: Corrosion
Results: There are about 580,000 publications during the latest 10 years, including 1065 "highly cited papers" and 35 "hot papers". Annual publications are shown in the histogram, where 2022 result is incomplete now and thus not shown. Figure S2. Experimental data of corrosion coverage (θ) and corrosion mass (Mcorro) of iron sheets in deionized water (DI) and the confirmation of our theoretical prediction.

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The data of θ was fitted with equation (6) Table S1. Ion and glucose concentrations (mol/L) of the aqueous corrosion media examined in this study.   Figure S4. Global views of corrosion surfaces of iron sheets after immersion in the Figure S10. SEM images of iron after immersion for 4 hours and 7 days in the indicated four media. On the iron surface after immersed in AS for 7 days, the loose corrosion products on the specimen dropped off when the specimen was taken out of the medium. S20 Table S5. Element contents in the blue boxes of Figure S10   S24 Figure S14. The corrosion rates calculated based on corrosion mass within 7 days.